Related papers: Evaluation modules for quantum toroidal ${\mathfra…

In this paper, we study the irreducible objects of the category Cf in of integrable representations for Map full Toroidal Lie algebras with finite dimensional weight spaces. These representations turn out to be single point evaluation…

We construct a representation of the affine W-algebra of gl_r on the equivariant homology space of the moduli space of U_r-instantons on A^2, and identify the corresponding module. As a corollary we give a proof of a version of the AGT…

We study the action of the quantum group $U_q(\widehat{\mathfrak{gl}_n})$ on the equivariant K-theory of affine Laumon spaces. We show that, at any highest weight away from the critical level, this can be identified with the contragredient…

A number of affine-Weyl-invariant integrable and exactly-solvable quantum models with trigonometric potentials is considered in the space of invariants (the space of orbits). These models are completely-integrable and admit extra particular…

We provide geometric constructions of modules over the graded Cherednik algebra $\mathfrak{H}^{gr}_\nu$ and the rational Cherednik algebra $\mathfrak{H}^{rat}_\nu$ attached to a simple algebraic group $\mathbb{G}$ together with a pinned…

The partition functions of a 2D conformal system - the modular invariant one or the generalized ones, coming from the introduction of defect lines - are expressed in terms of a set of coefficients that have the particularity to form nimreps…

The relaxed highest weight representations introduced by Feigin et al. are a class of representations of the affine Kac-Moody algebra $\hat{\mathfrak{sl}_2}$, which do not have a highest (or lowest) weight. We formulate a generalization of…

We introduce an algebra $\mathcal{K}_n$ which has a structure of a left comodule over the quantum toroidal algebra of type $A_{n-1}$. Algebra $\mathcal{K}_n$ is a higher rank generalization of $\mathcal{K}_1$, which provides a uniform…

We study Demazure modules which occur in a level $\ell$ irreducible integrable representation of an affine Lie algebra. We also assume that they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie…

Let $L((n-\tfrac 3 2)\Lambda_0)$, $n \in \Bbb N$, be a vertex operator algebra associated to the irreducible highest weight module $L((n-\tfrac 3 2)\Lambda_0)$ for a symplectic affine Lie algebra. We find a complete set of irreducible…

We study a class of representations over the degenerate double affine Hecke algebra of gl_n by an algebraic method. As fundamental objects in this class, we introduce certain induced modules and study some of their properties. In…

We consider the (direct sum over all $n$ of the) $K$-theory of the semi-nilpotent commuting variety of $\mathfrak{gl}_n$, and describe its convolution algebra structure in two ways: the first as an explicit shuffle algebra (i.e. a…

We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the…

We use the Hecke algebras of affine symmetric groups and their associated Schur algebras to construct a new algebra through a basis, and a set of generators and explicit multiplication formulas of basis elements by generators. We prove that…

We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the…

To each symmetric algebra we associate a family of algebras that we call quantum affine wreath algebras. These can be viewed both as symmetric algebra deformations of affine Hecke algebras of type $A$ and as quantum deformations of affine…

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra.…

We first prove that the K-theoretic Hall algebra of a preprojective algebra of affine type is isomorphic to the positive half of a quantum toroidal quantum group. An essential step consists to deform the K-theoretic Hall algebra so that the…

Let $R$ be a root datum with affine Weyl group $W^e$, and let $H = H (R,q)$ be an affine Hecke algebra with positive, possibly unequal, parameters $q$. Then $H$ is a deformation of the group algebra $\mathbb C [W^e]$, so it is natural to…

We classify extended Abelian Chern-Simons theories with gauge group $U(1)^n$ as extended $(2+1)$-dimensional topological quantum field theories. For an even integral nondegenerate lattice $(\Lambda,K)$, let $(G_K,q_K)$ denote its…